The General Use of the Canon and Tables of Logarithms.

Logarithmetick is a Logical kind of Arithmetick, or artificial use of Numbers invented for ease of the Calculation, wherein each Number is fitted with an Artificial, and these Artificial Numbers so ordered, that what is produced by Multiplication of Natural Numbers, the same may be effected by the Addition of these: their Artificial Numbers; what they perform by Division, the same is here done by Subtraction: and so the hardest part of Calculation avoided by an easie Probsthaphaeresis.

All this shall be made plain by applying that to these Artificial Numbers, which I have set down before, for the use of my Lines of Numbers, Sines and Tangents in the Use of the Sector and Cross-Staff. Wherein the Reader is to observe, that what is to be wrought by round Numbers only, is best done by Mr. Briggs his Logarithms, but the Astronomical part concerning Arks and Angles, by my Canon of Artificial Sines and Tangents.

Chapt. I.
Concerning the Use of the Line of Numbers,

I have set down ten general Proposition in the first Book of the Use of the Cross-Staff, Chapt. VI. and those may be applied to the Table of Logarithms.

Prop. I.
To multiply one Number by another.

This is the sixth Proposition of the ten; but I begin with the easiest, add the Logarithm of the Multiplicator, to the Logarithm of the Multiplicand, the Sum of both shall be the Logarithm of the Product.

And forward: All intermediate Numbers which have intermediate Logarithms.

If we multiply 101 by 10, the Product is 1010; of 102 by 10, the Product is 1020:

The Difference being only in the first Figure, and that is always less by one than the number of Places, in the Number given. As when we find the Logarithm to be 2,00860017 the first Figure 2, is Characteristical, i. e. the Index, shewing that the whole number 102 belonging to this Logarithm, consists of three places. If the Logarithm had been 1,00860017, the whole Number must have been 10.2 consisting of two places, and the rest a Fraction 2/10.

If the Logarithm were 0,00860017 the Number belonging to it would be 1.02, i. e. 1 and 02/100. And this is one of the reasons why the Differences were omitted in the first hundred Logarithms. All those Logarithms may be found afterwards under a larger Index.

Again, if we multiply 201 by 5, the Product is 1005: so here: If we add the Logarithm of 5 unto the Logarithm of 201, the Sum of both shall be the Logarithm of 1005, and the Sum of the Logarithms of 5 and 203 shall be the Logarithm of 1015. Thus the most part of the Table may be continued beyond 1000.

Prop. II.
To divide one Number by another.

Subtract the Logarithm of the Divisor, out of the Logarithm of the Dividend, the Remainder shall be the Logarithm of the Quotient.

Wherefore, if it were required to find the Logarithm of a whole Number with a Fraction annexed (as of 2 1/4), we might first reduce it into an improper Fraction of 11/4 (or rather 275/100) and then subtract as before.

If it were required to find the Logarithm of a single Fraction, as of 4/11, we may substract as before: But this Fraction being less than 1, the Logarithm must be less than 0, and there fore noted with - a defective sign.

PROP. IV.
To find the Cubique Root of a Number.

The third part of the Logarithm of the Number given, is the full Logarithm of the Cubique Root.

Then multiplying the Square and Cubique Root one by another, we may produce xxx other Number, and have all these Logarithms.

PROP. V.
Three Numbers being given, to find a fourth Proportional.

This Golden Rule the most useful of all others may be wrought several ways, as it appears by this Example.

As 12 unto 24: so 4 to a fourth number.

The ordinary way in Arithmetick is by Multiplication and Division. (I. Tacitus 2 & 3 divisus per 1.) For first they multiply the second into the third, and then divide the Product by the first Number given. As here, multiplying 24 by 4, the Product is 96, then dividing 96 by 12, the Quotient will be 8, the fourth number here required.

According to this way we add the Logarithm of the second and the third, and subtract the Logarithms of the first, so that which remaineth shall be the Logarithm of the fourth Number required.

A second way in Arithmetick is tby Division and Multiplication. For where the second Number is greater than the first, they may divide the second by the first, and then multiply the third by the Quotient. (II. Quotitents 2 per 1 divisi multiplicatus in tertiu.) As here, dividing 24 by 12, the Quotient is 2: then multiplying 4 by 2, the product will be 8.

According to this way we take the Logarithm of the first out of the Logarithm of the second, and then add the difference to the Logarithm of the third. So the Sum of this Addition shall be the Logarithm of the fourth required.

A third way in Arithmatick is by Division and Division, for where the second Number is less than the first, they may divide the first by the second, and then again divide the third by the Quotient. As here, dividing 12 by 4, the Quotient is 3: then dividing 24 by 3, the Quotient is 8. (III. Quotient xxx & divise 3.)

According to this way we take the Logarithm of the second out of the Logarithm of the first, and then take the Difference out of the Logarithm o the third: so that which remanieth shall be the Logarithm of the fourth Number required.

These two latter ways by Difference of Logarithms, may be considered as the same. Though there be some difference between them, yet that may easily be reconsidered, if we have regard to the nature of the question. For three members being given I direct proportion, if the second be greater than the first, the fourth must be greater than the third: If the second be less than the first, the fourth must be less than the third, and their Logarithms accordingly. But in reciprocal proportion, considering the first and second numbers to be one denomination, we are to observe the contrary.

If we desire to turn Subtraction into Addition, we may take the Logarithm which is to be subtracted out of the Radius, and add the Complement. So the Sum of this Addition, the Radius being subtracted, shall give the required Logarithm as before.

Thus in the last Example: where subtracting the Difference 4.7712126 out of 1.38021124, the Logarithm of 24, we found the Remainder to be 0.90308998 the Logarithm of 8.

From this, if we subtract the Radius, (that is, if we cancel the first figure to the left hand) the rest is 0.90308998 the Logarithm of 8, the fourth Proportional as before.

By help of this fourth Proportional we may come somewhat near to find a Logarithm for a number of 6 places.

As if it were required to find a Logarithm for this number 868624, the Table will afford us Logarithms for a lesser and a greater number; and then the intermediate may be found by the part proportional in this manner.

• As 1000 unto 50005
• So 624 unto 31203

The part proportional to be added to the lesser Logarithm 5.93851973 so shall we have 5.93883176 for the Logarithm required.

In like manner, having a Logarithm given, we may find the value of it in a number of fix places.

• As if the Logarithm given were 3.93882182
• And it were required to find the Number to which it belongeth:

The part proportional to be joined to the end of the former number 868: so shall we have 86861411 for the value of that Logarithm. But the Index of the Logarithm being 3, the Number required must consist of four places, viz. 8686, and the rest Fraction of 24/100.

This I say is somewhat near the Truth. For this number here proposed 868614 is the Square of 932.

PROP. VI.
Three Numbers being given, to find a fourth in a duplicated Proportion.

In Questions that hold in a duplicated Proportion between Lines and Superficies, the Logarithms for Lines may be doubled, the Logarithms for Lines required may be halfed, and then the work will be the same s in the first parts of the former Proposition.

Suppose, the Diameter being 14, the content of the Circle was 154, the Diameter being 28, what may the content be?

Here the Question concerns both Lines and Superficies, I double the Logarithm of the two Lines given, and then work as before in this manner:

Suppose the content of the Circle being 154, the Diameter of it was 14; the content being 616, what may the Diameter be?

Here being one Line given, and one Line required, I double the Logarithm of the Line given, and then working as before, the half of the remainder shall be the Logarithm of the Line required.

Or according to the second manner of operation, the difference between the Logarithms of Lines may be doubled; the difference between the Logarithms of the content given may be halfed, and then the work will be the same as in the latter part of the former Proposition.

So in the first Question, where the Diameters were given and the content required.

PROP. VII.
Three Numbers being given, to find a fourth in a triplicated Proportion.

In Questions concerning Proportion between Lines and Solids, the Logarithms for Lines given may be triplicated; The Logarithms for Lines required may be divided into three parts, and then the works will be the same, as in the first way for the Rule of Three.

Suppose the Diameter of an Iron Bullet being four in. has, the weight of it was nine pounds, [if the Diameter were 8] inches, what may the weight be?

Suppose the Weight of an Iron Bullet being nine pound, the Diameter was four inches; the Weight being seventy two pound, what may the Diameter be?

Or according unto the second manner of operation in the Rule of Three, the Difference between Logarithms of Lines given may be tripled; the Difference between Logarithms of the Solidity or Weight given may be divided into three parts.

So in the first Question, where the Diameters were known, and the Weight required.

In the second Question, where the Weight was known, and the Diameter required.

PROP. VIII.
Having two Numbers given, to find a third in continual Proportion, a fourth, a fifth, a sixth, and so forward.

According to the first way in the Rule of Three, we may subtract the Logarithm of the first number, out of double the Logarithm of the second, the remainder shall be the Logarithm of the third, then subtraction the Logarithm of the first Number again, out of the Logarithms of the second and third, that is, out of triple the Logarithm of the second, the remainder shall be the Logarithm of the fourth, and so forward.

As, when we say: as 1 unto 2, so 2 unto 4, and 4 unto 8, and 8 unto 16 &c. Because the first Number is 1, there is no need of Division, but only to multiply a second Number into it self, the Product gives the third Proportional Number to be 4: then multiplying 2 into 4, the fourth Proportional is 8: and multiplying 2 into 8, the fifth Proportional is 16; and so forward. So here the Logarithm of the first number being 1, there is no need of Subtraction.

In all other numbers that begin not with 1, we may either subtract the Logarithm of the first Number or add the Complement unto the Radius.

As when the Numbers given are 100 and 108.

According to the second manner of Operation we may take the Difference between the Logarithms of two Numbers given; so this Difference applied to the Logarithm of the second Number, shall give the Logarithm of the third Proportional: the same Difference applied to Logarithm of the third Proportional, shall give the Logarithms of the fourth Proportional, or double of this Difference applied to the Logarithm of the first Number, shall give the Logarithm of the third Proportional: the treble of this Difference applied to the Logarithm of the first Number, shall give the Logarithm of the fourth Proportional; and so forward.

As in the former Example where the two Numbers given were 100 and 108, suppose 100 increasing to 108, and so yearly in continual Proportion after the Rate of 8 in the 100, and that it were required to find what this 100 would grow unto by the end of 20 years.

That is 466 l. 1 s. 14 d. fere, the Sum that 100 would grow unto by the end of 20 years at the rate proposed.

In like manner if the two first Numbers given were 108 and 100: Suppose 108 decreasing to the 100, and so yearly in continual proportion and that it were required to find what 100 would decrease unto by the end of 20 years: Or (which is all one (suppose 100 to be due 20 years hence, and that it were required to find the worth thereof in ready money according to the former rate.

the Logarithm of 21 4548/10000 that is 21 l. 9 s. 1 d. and such is the present worth of 100 l. due at the end of 20 years; So that this present worth being taken forth of the 100 l. principal debt, here remains 78 l. 10 s. 11 d. for the present worth of the continued gain that may be made either of the loan of 100 l. or of 8 l. Annuity after 20 years according to the former rate.

If a Lease of 100 l. by the year, or such other yearly Pension were to continue for 20 years, and that it were required to find the worth thereof in ready money. This might be found upon the same ground of continual proportion, and that several ways.

1. It appeareth before, that 100 l. due at the years end is worth but 92 124/100 in ready money: If it be due at the end of two years, the present worth is 85 l. 111/100: then adding these two together, we have 178 l. 124/100 for the present worth of 100 l. Annuity for two years, and so forward.
2. It appeareth before that the present worth of 8 l. for 20 years is 78 l 5452: and then it follows by proportion.

   As an Annuity of	8 l. 000	0.90308999
   Is to the worth thereof	981.8147	2.99202954

3. 3. As the yearly Loan of 100 l. includes an Annuity of 8 l. So there is a Sum equivalent to 100 l. Annuity.

This Sum equivalent may be diminished according to the Number of years as before: to the Complement of the Sum diminished to the Sum equivalent shall be the present worth of the Annuity.

Whose Complement to 1250 is 981.8147, that is 981 l. 16 s. 3 d. ob. and such is the present worth of 100 l. Annuity for 20 years, at the rate of 8 in the 100 per annum.

The like reason holdeth for any other rate and time proposed.

PROP. IX.
Having two extreme Numbers given, to find a mean Proportional between them.

Add the Logarithms of the two extreme Numbers: the one half of the Sum shall be the Logarithm of the mean proportional.

PROP. X.
Having two extreme Numbers given, to find two mean Proportionals between them.

In the ordinary way of Arithmetick we commonly multiply the greater Extreme by the Square of the lesser, so the Cubique Root of the Product shall be the lesser mean: then multiply the lesser Mean into the greater Extreme, the Square Root of the Product shall be the greater Mean Proportional: Or having found the lesser Mean, we may find the other Mean by continual Proportion.

Accordingly we may add the Logarithm of the greater Extreme, to double the Logarithm of the lesser, so the third part of the Sum shall be the Logarithm of the lesser Mean. Then adding this Logarithm of the lesser Mean, to the Logarithm of the greater Extreme, the one half of the Sum shall be the Logarithm of the greater Mean Proportional.

Or according to the second manner of Operation in the Rule of Three, (which is the work that I always follow in the Line of Numbers) we may take the Difference between the Logarithms of the two extreme Numbers, and divide this Difference into three equal parts, so the Sum of the Logarithm of the lesser Extreme and 1/3 part, shall be the Logarithm of the lesser Mean: the Sum of this Logarithm of the lesser Mean and the same 1/3 part, shall be the Logarithm of the Greater Mean Proportional.

And by the same reason, if it were required to find the three Mean Proportionals, we might divide the former Difference into four equal part and so forward.

As if it were required to find the first od eleven mean Proportional between 100 and 108. Or (which is all one) suppose 100 l. increasing in continual Proportion, so as that by the end of 12 months it came to 108 l. and that it were required to find what this 100 l. did grow by the end of the first Month.

Then having these two, 100 and 100.64340301, together with 108, the last of the twelfe, the other Intermediate may be found by continual Proportion as before.

This Explication of my ten former propositions may serve for the frugal Use of the Table of Logarithms. Those which require more may have recourse to the Treatise which is mentioned before in the Front of the Table.